Mathematische Annalen

, Volume 355, Issue 4, pp 1383-1423

\(C^*\)-algebras of Toeplitz type associated with algebraic number fields

  • Joachim CuntzAffiliated withMathematisches Institut Email author 
  • , Christopher DeningerAffiliated withMathematisches Institut
  • , Marcelo LacaAffiliated withMathematics and Statistics, University of Victoria

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We associate with the ring \(R\) of algebraic integers in a number field a C*-algebra \({\mathfrak T }[R]\). It is an extension of the ring C*-algebra \({\mathfrak A }[R]\) studied previously by the first named author in collaboration with X. Li. In contrast to \({\mathfrak A }[R]\), it is functorial under homomorphisms of rings. It can also be defined using the left regular representation of the \(ax+b\)-semigroup \(R\rtimes R^\times \) on \(\ell ^2 (R\rtimes R^\times )\). The algebra \({\mathfrak T }[R]\) carries a natural one-parameter automorphism group \((\sigma _t)_{t\in {\mathbb R }}\). We determine its KMS-structure. The technical difficulties that we encounter are due to the presence of the class group in the case where \(R\) is not a principal ideal domain. In that case, for a fixed large inverse temperature, the simplex of KMS-states splits over the class group. The “partition functions” are partial Dedekind \(\zeta \)-functions. We prove a result characterizing the asymptotic behavior of quotients of such partial \(\zeta \)-functions, which we then use to show uniqueness of the \(\beta \)-KMS state for each inverse temperature \(\beta \in (1,2]\).

Mathematics Subject Classification (2000)

Primary 22D25 46L89 11R04 11M55